Understanding the Sequence and the Median

The question asks for the median of the sequence 2, 4, 6, ..., 100. This sequence consists of all even numbers starting from 2 up to 100.

This is an arithmetic progression where:

• The first term (a) is 2.
• The common difference (d) is 4 - 2 = 2 (the difference between consecutive terms).
• The last term (\(a_n\)) is 100.

Finding the Number of Terms in the Sequence

To find the median of a sequence, we first need to know the total number of terms. For an arithmetic progression, we can use the formula for the n-th term:

\(a_n = a + (n-1)d\)

We know \(a_n = 100\), \(a = 2\), and \(d = 2\). Let's plug these values into the formula:

\(100 = 2 + (n-1)2\)

Subtract 2 from both sides:

\(100 - 2 = (n-1)2\)

\(98 = (n-1)2\)

Divide both sides by 2:

\(\frac{98}{2} = n-1\)

\(49 = n-1\)

Add 1 to both sides:

\(n = 49 + 1\)

\(n = 50\)

So, there are 50 terms in the sequence 2, 4, 6, ..., 100.

Calculating the Median for an Even Number of Terms

The median is the middle value of a dataset when it is arranged in ascending order. Since our sequence is already in ascending order (2, 4, 6, ..., 100) and we have an even number of terms (n=50), the median is the average of the two middle terms.

The positions of the two middle terms are:

• The \( \frac{n}{2} \)th term
• The \( (\frac{n}{2} + 1) \)th term

In this case, \( n = 50 \), so the middle terms are at positions:

• \( \frac{50}{2} = 25 \)th term
• \( (\frac{50}{2} + 1) = (25 + 1) = 26 \)th term

We need to find the value of the 25th term (\(a_{25}\)) and the 26th term (\(a_{26}\)) in the sequence.

Using the formula \(a_n = a + (n-1)d\):

For the 25th term (\(a_{25}\)):

\(a_{25} = 2 + (25-1)2 = 2 + (24)2 = 2 + 48 = 50\)

For the 26th term (\(a_{26}\)):

\(a_{26} = 2 + (26-1)2 = 2 + (25)2 = 2 + 50 = 52\)

The two middle terms are 50 and 52.

Final Median Calculation

The median is the average of the 25th term and the 26th term:

\(\text{Median} = \frac{a_{25} + a_{26}}{2}\)

\(\text{Median} = \frac{50 + 52}{2}\)

\(\text{Median} = \frac{102}{2}\)

\(\text{Median} = 51\)

Therefore, the median of the sequence 2, 4, 6, ..., 100 is 51.

Revision Table: Key Concepts for Finding Median

Here's a quick review of how to find the median based on the number of terms:

Additional Information: Arithmetic Sequences and Median in Statistics

The sequence 2, 4, 6, ..., 100 is a perfect example of an arithmetic sequence. An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference.

The median is a measure of central tendency in statistics. It represents the middle value that separates the higher half from the lower half of a data set. Unlike the mean, the median is not affected by extreme outliers, making it a useful statistic for skewed distributions.

In general, to find the median of any dataset, you must first arrange the data points in ascending order. Then, apply the rules for finding the median based on whether the total count of data points is odd or even.